FAPT: a Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory

TL;DR

This paper introduces a comprehensive Mathematica package that facilitates calculations of analytic images of the strong coupling constant in QCD's Fractional Analytic Perturbation Theory, covering various energy scales and schemes.

Contribution

It provides all necessary procedures in Mathematica for computing analytic images of QCD coupling powers in Minkowski and Euclidean domains, including heavy-quark thresholds, for the first time.

Findings

Enables calculations of singularity-free QCD couplings at arbitrary scales.

Supports fixed and global schemes with heavy-quark thresholds.

Facilitates applications of Fractional Analytic Perturbation Theory in QCD analyses.

Abstract

We provide here all the procedures in \texttt{Mathematica} which are needed for the computation of the analytic images of the strong coupling constant powers in Minkowski (${\bar{\mathfrak A}_{\nu}(s;n_f)}$ and ${\mathfrak A_{\nu}^\text{glob}(s)}$) and Euclidean (${\bar{\mathcal A}_{\nu}(Q^2;n_f)}$ and ${\mathcal A_{\nu}^\text{glob}(Q^2)}$) domains at arbitrary energy scales (${s}$ and ${Q^2}$, correspondingly) for both schemes --- with fixed number of active flavours ${n_f=3, 4, 5, 6}$ and the global one with taking into account all heavy-quark thresholds. These singularity-free couplings are inevitable elements of Analytic Perturbation Theory (APT) in QCD and its generalization --- Fractional APT, needed to apply the APT imperative for renormalization-group improved hadronic observables.